Practice Maths

Arc Length and Sector Area

Key Ideas

Key Terms

sector
a "pie slice" region bounded by two radii and the arc between them; its angle θ is at the centre.
arc
the curved part of the circle's edge forming one side of a sector.
arc length
the length of the curved arc of a sector; arc length = (θ ÷ 360) × 2πr.
sector area
the area of the "pie slice" region; sector area = (θ ÷ 360) × πr².
angle (θ)
the central angle of the sector in degrees; determines what fraction of the full circle the sector represents: θ ÷ 360.
fraction of a circle
the proportion of the full circle that a sector occupies; fraction = θ ÷ 360. Used to scale both the circumference (for arc length) and area.
Hot Tip Think of arc length and sector area as fractions of the whole circle. If the angle is 90°, the sector is ¼ of the circle. If the angle is 180°, it is half the circle. This is a great way to check your answers make sense.

Worked Example

Question: A sector has radius 8 cm and angle 90°. Find the arc length and sector area. Round to 2 decimal places.

Step 1 — Find the fraction.
Fraction = 90 ÷ 360 = ¼

Step 2 — Find arc length.
Arc length = ¼ × 2π × 8 = ¼ × 16π = 4π ≈ 12.57 cm

Step 3 — Find sector area.
Sector area = ¼ × π × 8² = ¼ × 64π = 16π ≈ 50.27 cm²

The Key Idea: A Fraction of a Full Circle

An arc is a curved part of a circle's edge, and a sector is the "pie slice" shape cut out by two radii. Both are simply a fraction of the full circle, and that fraction is determined by the angle θ (theta) at the centre.

A full circle has an angle of 360°. If a sector has an angle of, say, 90°, it takes up 90360 = ¼ of the circle. If the angle is 180°, it is a semicircle — half the circle. The fraction θ360 is the single most important concept in this lesson.

Arc Length Formula

The arc length is the fraction θ360 of the full circumference (2πr):

Arc length = (θ ÷ 360) × 2πr

Example: Find the arc length of a sector with radius 8 cm and angle 45°.
Arc length = (45 ÷ 360) × 2 × π × 8
= (18) × 16π
= 2π
≈ 6.28 cm

Notice that 45° is exactly 18 of 360°, so the arc is 18 of the circumference — a nice check on the answer.

Sector Area Formula

The sector area is the fraction θ360 of the full circle area (πr2):

Sector area = (θ ÷ 360) × πr2

Example: Find the area of a sector with radius 10 cm and angle 120°.
Sector area = (120 ÷ 360) × π × 102
= (13) × 100π
= 100π3
≈ 104.72 cm2

Key tip: Both formulas follow the same pattern — multiply the fraction (θ360) by the full-circle version. Write out the fraction first, simplify it if possible (e.g. 90÷360 = ¼), then multiply. This reduces arithmetic errors and helps you spot mistakes quickly.

The Perimeter of a Sector

A common exam question asks for the perimeter of a sector — the total distance around the outside. Be careful: a sector's perimeter has three parts, not just the arc.

Perimeter of sector = arc length + radius + radius = arc length + 2r

Example: Using the 45°, r = 8 cm sector from above:
Arc length ≈ 6.28 cm
Perimeter = 6.28 + 8 + 8 = 22.28 cm

Mixed Problems and Reverse Questions

Some questions give you the arc length or sector area and ask you to find the angle or radius. Just substitute what you know and solve for the unknown.

Example — find the angle: A sector with r = 6 cm has an arc length of 4π cm. Find θ.
4π = (θ360) × 2π × 6
4π = (θ360) × 12π
412 = θ360
13 = θ360
θ = 120°

Always write your formula first, substitute the known values, and then solve step by step. This structured approach earns full working-out marks.

Mastery Practice

  1. Write each sector angle as a simplified fraction of a full circle. Fluency

    1. 90°
    2. 180°
    3. 60°
    4. 270°
    5. 120°
    6. 45°
    7. 30°
    8. 240°

  2. Find the arc length for each sector. Round to 2 decimal places. Fluency

    1. r = 5 cm, θ = 90°
    2. r = 10 m, θ = 180°
    3. r = 6 cm, θ = 60°
    4. r = 8 m, θ = 45°
    5. r = 12 cm, θ = 120°
    6. r = 4 m, θ = 270°
    7. r = 3 cm, θ = 30°
    8. r = 15 mm, θ = 144°

  3. Find the sector area for each sector. Round to 2 decimal places. Fluency

    1. r = 4 cm, θ = 90°
    2. r = 7 m, θ = 180°
    3. r = 6 cm, θ = 60°
    4. r = 10 m, θ = 45°
    5. r = 5 cm, θ = 270°
    6. r = 9 mm, θ = 120°
    7. r = 2 cm, θ = 30°
    8. r = 8 m, θ = 150°

  4. Find the perimeter of each sector (arc length + 2 radii). Round to 2 decimal places. Understanding

    1. r = 6 cm, θ = 90°
    2. r = 10 m, θ = 60°
    3. r = 5 cm, θ = 120°
    4. r = 8 mm, θ = 270°
    5. r = 3 m, θ = 45°
    6. r = 12 cm, θ = 30°

  5. Use the given information to find the unknown. Understanding

    1. A sector with r = 6 cm has arc length 6π cm. Find the angle θ.
    2. A sector with θ = 120° has sector area 12π cm². Find the radius.
    3. A sector with r = 10 m has sector area 25π m². Find the angle θ.
    4. A semicircle has perimeter 36.28 cm. Find the radius (to 2 dp).

  6. Arc length and sector area in real-world contexts. Problem Solving

    1. A clock has hands 12 cm long. How far does the tip of the minute hand travel in 20 minutes? Round to 2 decimal places.
    2. A pie is cut into 8 equal slices. The pie has a radius of 15 cm.
      1. What is the angle of each slice?
      2. Find the arc length of one slice.
      3. Find the area of one slice.
    3. A sector-shaped garden has radius 5 m and angle 150°. Turf costs $12 per m². Find the cost to turf the sector (to the nearest dollar).
    4. Two sectors have the same area. Sector A has radius 6 cm and angle 90°. Sector B has radius 9 cm. Find the angle of Sector B.

  7. Find the arc length and sector area for each sector shown. Round to 2 d.p. Understanding

    1. O r = 8 cm 90°
    2. O r = 6 m 120°

  8. Complete the table. Round to 2 d.p. Understanding

    Radius (r)Angle (θ)Fraction of circleArc lengthSector areaPerimeter of sector
    5 cm90°____________
    10 m180°____________
    8 cm60°____________
    12 mm270°____________
    4 m45°____________

  9. Each calculation has an error. Find it and correct it. Understanding

    1. For a sector with r = 10 cm and θ = 90°, a student found arc length = 90 ÷ 360 × π × 10² = 78.54 cm. What formula did they use incorrectly?
    2. For a sector with r = 6 m and θ = 60°, a student found sector area = 60 × π × 6² = 6786 m². What did they forget to do?
    3. A student said the perimeter of a sector with r = 5 cm and θ = 90° is just the arc length = 7.85 cm. What is missing from the perimeter?
    4. A student found the arc length of a semicircle with radius 7 m as arc = πr = π × 7 ≈ 21.99 m. Is this correct? Explain why or why not.

  10. Extended response. Show full working including the fraction of a circle at each step. Problem Solving

    1. A sprinkler rotates through an angle of 135° and waters a circular sector of radius 8 m.
      1. Find the arc length of the sector (to 2 dp).
      2. Find the area of the sector (to 2 dp).
      3. Lawn seed costs $6 per m². Find the total cost to seed the watered area.
    2. A wind turbine blade is 40 m long. As the blade rotates, the tip sweeps out a circle.
      1. The blade completes one full revolution. Find the distance the tip travels (to 2 dp).
      2. In half a revolution, find the arc length swept by the tip.
      3. Find the area swept by the blade in one full revolution.
    3. Two sectors are cut from the same circle of radius 10 cm. Sector A has angle 72° and sector B has angle 108°.
      1. Find the arc length of each sector.
      2. Find the area of each sector.
      3. Together, what fraction of the whole circle do sectors A and B make up? Confirm by adding their areas and comparing to the circle area.
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